Particle decay rates

Every unstable particle carries a clock and a menu. The clock is its mean lifetime $\tau$ ; the menu is the list of ways it can fall apart, each with its own probability. Particle physics turns both into one calculated number — the decay rate $\Gamma$ , the transition probability per unit time — through the relativistic form of Fermi's golden rule. The rate of any single process is the product of two utterly different things: a dynamical matrix element $\mathcal{M}_{fi}$ that encodes which interaction does the job and how strongly, and a kinematical phase-space integral that merely counts how much room the final-state particles have to fly into. This factorization is the organizing idea of the whole subject: the same phase-space machinery serves a pion, a muon, a $Z$ boson, and a Higgs, while only $\mathcal{M}_{fi}$ changes from one to the next. A particle with several open channels has a partial rate $\Gamma_j$ for each. Because distinct channels are independent, the total rate is their sum, $\Gamma=\sum_j\Gamma_j$ , and this total — not any single channel — sets the lifetime through $\tau=\hbar/\Gamma$ . An ensemble then depletes exponentially, $N(t)=N(0)e^{-\Gamma t}$ . The fraction going through a given channel is its branching ratio $\mathrm{BR}_j=\Gamma_j/\Gamma$ , and the branching ratios sum to one by construction. For the ubiquitous two-body decay the phase-space integral can be done once and for all: the final momentum in the rest frame, $p^*$ , is fixed entirely by the three masses, and the rate collapses to a single tidy formula.

Key takeaways

A decay rate is $\Gamma$ = (matrix element squared) × (phase space): dynamics and kinematics factorize cleanly.
The total width $\Gamma=\sum_j\Gamma_j$ sums the partial widths; the lifetime is $\tau=\hbar/\Gamma$ and survival is $N(t)=N(0)e^{-\Gamma t}$ .
The branching ratio $\mathrm{BR}_j=\Gamma_j/\Gamma$ is the fraction decaying through channel $j$ ; all branching ratios sum to unity.
For two bodies the phase space is universal: $\Gamma=\dfrac{p^*}{32\pi^2 m_a^2}\displaystyle\int|\mathcal{M}_{fi}|^2\,d\Omega$ , with $p^*$ fixed by the masses alone.
Width and lifetime are the same information: a narrow width means a long life ( $\Gamma_Z=2.50$ GeV gives $\tau_Z=2.6\times10^{-25}$ s; $\tau_\pi=2.6\times10^{-8}$ s gives $\Gamma_\pi=25$ neV).
Key idea: A decay rate splits cleanly into 'what interaction does it' (the matrix element) and 'how much room is there' (phase space); for two bodies the phase space collapses to a single momentum $p^*$ fixed by the masses.

Where this lives today

Decay rates are the daily currency of particle physics, and the Particle Data Group's tables are, at bottom, one giant catalogue of widths and branching ratios. The most celebrated application remains the $Z$ lineshape at LEP: measuring the total width of the $Z$ resonance to a few parts in $10^4$ , and subtracting the visible (hadronic and charged-leptonic) partial widths, left an invisible width that counts light neutrino species — the answer, $N_\nu=2.984\pm0.008$ , told us there are exactly three generations of light neutrinos, a result no accelerator built to make neutrinos could have reached so cleanly. The same width-measurement logic drives flavour physics: the parts-per-thousand differences among $B$ -meson lifetimes and the tiny $CP$ -violating rate asymmetries at LHCb and Belle II are decay rates measured to extraordinary precision to test the CKM picture. At the frontier sits the Higgs boson, whose predicted total width of about 4 MeV is far too narrow to resolve directly at the LHC; experimenters instead infer it from the ratio of on-shell to off-shell production rates — a beautiful indirect use of the branching-ratio formalism. And every neutrino-oscillation and proton-decay search is, in the end, a decay-rate measurement pushed to its statistical limit, whether hunting a rate that is astonishingly slow (the proton lifetime bound now exceeds $10^{34}$ years) or reconstructing the rate of a state that never lived long enough to leave a track.

The physical picture

A particle at rest is a bomb with a probability, not a timer. Quantum mechanics gives no moment of detonation — only a constant rate $\Gamma$ that a decay will happen in the next instant, independent of how long the particle has already lived. That single number, computed from the interaction, controls everything downstream: the lifetime is its reciprocal, the survival curve is its exponential, and the split among competing modes is the ratio of the rates. The first figure freezes the moment of a two-body decay in the rest frame, where momentum conservation forces the products out back-to-back with equal and opposite momentum $p^*$ — a quantity set, remarkably, by nothing but the three masses.

Figure

Two-body decay in the parent rest frame: the products fly back-to-back with equal and opposite momenta of magnitude p, fixed by the masses alone. For π⁺ → μ⁺ ν_μ with m_π = 139.57 MeV and m_μ = 105.66 MeV, p = (m_π² − m_μ²)/(2m_π) = 29.8 MeV/c

Once the rate $\Gamma$ is fixed, the fate of a population follows without any further physics. Because the per-particle probability of decaying in the next instant is constant, an ensemble of $N$ particles loses a fixed fraction per unit time, $dN/dt=-\Gamma N$ , and the solution is a pure exponential. There is no memory: a muon that has survived a microsecond is exactly as likely to decay in the next instant as a freshly made one. The second figure is that survival curve, and it makes the meaning of the mean lifetime $\tau=1/\Gamma$ concrete — it is the time at which the population has fallen to $1/e\approx0.368$ of its start, drawn here for the muon.

Figure

Exponential survival N(t)/N(0) = e^(−t/τ) for the muon, τ = 2.197 μs. The population falls to 1/e = 0.368 after one mean lifetime and to 1/e² = 0.135 after two — a memoryless law fixed entirely by the total rate Γ = 1/τ

The core equations

The starting point is the relativistic golden rule. It gives the differential rate to scatter a parent of mass $m_a$ into a particular final state as the squared amplitude times a Lorentz-invariant measure that enforces four-momentum conservation and counts states. The leading $1/2m_a$ is the relativistic normalization of the initial particle; the delta function is energy–momentum conservation; the product runs over each final particle's invariant phase-space element.

d\Gamma = \frac{1}{2m_a}\,\overline{|\mathcal{M}_{fi}|^2}\,(2\pi)^4\,\delta^4\!\Big(p_a-\textstyle\sum_f p_f\Big)\,\prod_f \frac{d^3\mathbf{p}_f}{(2\pi)^3\,2E_f} \qquad (1)

where: $m_a$ — parent mass; $\overline{|\mathcal{M}_{fi}|^2}$ — spin-averaged (initial) and spin-summed (final) matrix element squared; $\delta^4$ — energy–momentum conservation; $d^3\mathbf{p}_f/((2\pi)^3 2E_f)$ — the invariant phase-space element of final particle $f$ . Reading it: rate = dynamics ( $|\mathcal{M}|^2$ ) × kinematics (everything else). Carrying out the two-body phase-space integral of (1) — the delta functions eat two of the six final momentum components and fix the magnitude of the rest — leaves a formula of striking economy. All of the kinematics has boiled down to one number, the common rest-frame momentum $p^*$ , sitting in front of an angular integral of the matrix element.

\Gamma_{a\to 1+2} = \frac{p^*}{32\pi^2\,m_a^2}\int |\mathcal{M}_{fi}|^2\,d\Omega \qquad (2)

where: $p^*$ — magnitude of either final momentum in the rest frame; $m_a$ — parent mass; $d\Omega$ — solid angle of one product. If $|\mathcal{M}_{fi}|^2$ is independent of angle (as for a spin-0 parent), the integral is just $4\pi|\mathcal{M}|^2$ and $\Gamma=p^*|\mathcal{M}|^2/(8\pi m_a^2)$ . The single kinematic input $p^*$ is fixed by the three masses through the Källén "triangle" function. It is symmetric in the two daughter masses and vanishes at threshold, when the parent is only just heavy enough to make the products at rest.

p^* = \frac{1}{2m_a}\sqrt{\big[m_a^2-(m_1+m_2)^2\big]\big[m_a^2-(m_1-m_2)^2\big]} \qquad (3)

where: $m_1,m_2$ — daughter masses; the bracketed product is the Källén function $\lambda(m_a^2,m_1^2,m_2^2)$ . Reading it: $p^*\to 0$ as $m_1+m_2\to m_a$ (no phase space), and $p^*\to m_a/2$ when both daughters are massless (maximal phase space). When several channels are open they add. Because the channels are independent quantum processes, their rates sum into a total width that alone sets the lifetime, while each channel's share is its branching ratio.

\Gamma=\sum_j \Gamma_j,\qquad \tau=\frac{\hbar}{\Gamma},\qquad \mathrm{BR}_j=\frac{\Gamma_j}{\Gamma},\qquad \sum_j \mathrm{BR}_j=1 \qquad (4)

where: $\Gamma_j$ — partial width of channel $j$ ; $\Gamma$ — total width; $\tau$ — mean lifetime; $\mathrm{BR}_j$ — branching ratio. Note $\hbar=6.582\times10^{-25}\ \mathrm{GeV\cdot s}$ ; in natural units ( $\hbar=1$ ) simply $\tau=1/\Gamma$ . Finally the total rate governs the whole population. A constant per-particle probability yields a first-order equation whose solution is the pure exponential survival law that defines the lifetime operationally.

\frac{dN}{dt}=-\Gamma N \;\Longrightarrow\; N(t)=N(0)\,e^{-\Gamma t}=N(0)\,e^{-t/\tau} \qquad (5)

where: $N(t)$ — number surviving at time $t$ ; $\Gamma$ — total decay rate; $\tau=1/\Gamma$ — mean lifetime, the $1/e$ point. The half-life is $t_{1/2}=\tau\ln 2\approx0.693\,\tau$ .

Derivation AFrom the golden rule to the invariant decay rate

The goal is to see precisely how the relativistic golden rule (1) arises and why the rate is unavoidably a product of a matrix element and a Lorentz-invariant phase space. We start from time-dependent perturbation theory, where the transition probability per unit time is proportional to the squared coupling times the density of accessible final states, and we then dress each factor with the relativistic normalizations that make every piece Lorentz invariant, so that a rate computed in one frame boosts correctly to any other. The payoff is a clean separation: the physics of the interaction lives entirely in $\mathcal{M}_{fi}$ , and the counting of final states lives entirely in the phase-space measure.

Step 1. Non-relativistic Fermi's golden rule reads $\Gamma = 2\pi|T_{fi}|^2\,\rho(E_f)$ : the rate is the squared transition matrix element times the density of final states at the energy that conserves energy. Why: perturbation theory says a weak coupling drives transitions at a rate set by how strongly it connects initial and final states and by how many final states are available to receive the particle.

Step 2. Pass to a relativistic, covariant normalization. Amplitudes are written with states normalized to $2E$ particles per unit volume rather than one, which replaces $T_{fi}$ by the invariant amplitude $\mathcal{M}_{fi}$ and attaches a factor $1/(2E)$ to every external leg. Why: the naive one-particle-per-box normalization is not Lorentz invariant — a boosted box is length-contracted — so the covariant scheme absorbs the boost factors into the states, leaving $\mathcal{M}_{fi}$ frame independent.

Step 3. Replace the crude density of states by the invariant phase-space element for each final particle, $d^3\mathbf{p}_f/[(2\pi)^3\,2E_f]$ , and enforce total four-momentum conservation with $(2\pi)^4\delta^4(p_a-\sum p_f)$ . Dividing by the initial-state normalization $2m_a$ (the parent is at rest) assembles equation (1). Why: $d^3\mathbf{p}/(2E)$ is the unique Lorentz-invariant measure on the mass shell, so writing the density of states this way keeps the whole rate invariant while still counting exactly the physical final states.

Step 4. Read off the structure: $d\Gamma = (\text{dynamics})\times(\text{kinematics})$ , with $\overline{|\mathcal{M}_{fi}|^2}$ the only process-specific piece and the delta function plus measure a universal template that depends only on how many particles are in the final state. Why: this is exactly the factorization the whole subject exploits — compute the phase space once per final-state multiplicity, and reuse it for every interaction that shares that multiplicity. *Note: equation (1) reduces to the non-relativistic golden rule in the low-energy limit, and every factor carries the correct mass dimension so that $\Gamma$ has dimension of energy.*

Go deeper · Derivation B · 1 minCollapsing the two-body phase space to a single momentum

Derivation BCollapsing the two-body phase space to a single momentum

The aim is to perform the phase-space integral in (1) for a two-particle final state and thereby derive the master formula (2) together with the Källén expression (3) for $p^*$ . The two-body case is special because energy and momentum conservation are just enough to freeze every kinematic degree of freedom except the overall orientation: once the parent mass is fixed, the daughters must emerge back-to-back with a single determined momentum magnitude, so the six-dimensional integral collapses to an integral over solid angle alone. This is why a two-body rate is always "one momentum times an angular average of the matrix element."

Step 1. In the rest frame the momentum delta function forces $\mathbf{p}_2=-\mathbf{p}_1$ , so both particles share a common momentum magnitude $p^*\equiv|\mathbf{p}_1|=|\mathbf{p}_2|$ . This kills the $d^3\mathbf{p}_2$ integral entirely. Why: with the parent at rest there is zero total momentum, and momentum conservation is the strongest possible constraint — it ties the two daughters rigidly together.

Step 2. The remaining $d^3\mathbf{p}_1 = p_1^2\,dp_1\,d\Omega$ meets the energy delta function $\delta(m_a-E_1-E_2)$ with $E_i=\sqrt{p_1^2+m_i^2}$ . Integrating over $p_1$ uses the rule $\int dp_1\,\delta(f(p_1)) = 1/|f'(p^*)|$ , and $|dE/dp_1| = p_1(1/E_1+1/E_2) = p^*\,m_a/(E_1E_2)$ evaluated at the root. Why: the energy constraint has a single solution $p_1=p^*$ , and the Jacobian of the delta function converts "integrate over the shell" into "evaluate at the shell, divided by the slope."

Step 3. Assembling the surviving factors — the two $1/(2E_i)$ normalizations, the $(2\pi)$ powers, the Jacobian, and the $1/2m_a$ — the energy factors cancel cleanly and leave equation (2), $d\Gamma = |\mathcal{M}_{fi}|^2\,p^*\,d\Omega/(32\pi^2 m_a^2)$ . Why: the miracle of the two-body case is that all the $E_1,E_2$ dependence introduced by the normalizations is exactly undone by the Jacobian, leaving only $p^*$ and $m_a$ .

Step 4. Solve the energy constraint $m_a=\sqrt{p^{*2}+m_1^2}+\sqrt{p^{*2}+m_2^2}$ for $p^*$ . Isolating and squaring twice gives $p^* = \tfrac{1}{2m_a}\sqrt{[m_a^2-(m_1+m_2)^2][m_a^2-(m_1-m_2)^2]}$ , the Källén result (3). Why: this is pure algebra of the mass shell, and it shows $p^*$ depends on nothing but the three masses — the daughters cannot choose their speed, only their direction. *Note: for $m_2=0$ the formula gives $p^*=(m_a^2-m_1^2)/2m_a$ ; for π⁺ → μ⁺ν this is $(139.57^2-105.66^2)/(2\times139.57)=29.79$ MeV.*

Go deeper · Derivation C · 2 minWhy decay is exponential and widths add

Derivation CWhy decay is exponential and widths add

The purpose here is to derive the survival law (5) and the additivity of partial widths (4) from a single physical premise — that the decay probability per unit time is a constant $\Gamma$ set by the golden rule and independent of the particle's history. From this one assumption the exponential law, the meaning of the mean lifetime, and the rule that competing channels simply add their rates all follow, and we can see exactly why the lifetime is governed by the total width rather than by any individual mode. The argument is short but it underlies every lifetime measurement ever made.

Step 1. For a large ensemble of $N$ identical particles, the expected number decaying in a small interval $dt$ is $\Gamma N\,dt$ , so $dN = -\Gamma N\,dt$ . Integrating gives $N(t)=N(0)e^{-\Gamma t}$ . Why: a constant per-particle rate means the ensemble loses a fixed fraction per unit time, and constant fractional loss is the definition of exponential decay.

Step 2. The mean lifetime is the probability-weighted average time to decay, $\tau=\int_0^\infty t\,\Gamma e^{-\Gamma t}\,dt = 1/\Gamma$ , so $\Gamma$ and $\tau$ are reciprocals; the survival curve reaches $1/e$ precisely at $t=\tau$ . Why: this identifies the abstract rate with the measurable average lifetime, and shows why $\tau$ is the natural time scale of the exponential rather than, say, the half-life.

Step 3. With several open channels, the total probability of decaying (through any channel) in $dt$ is the sum of the independent channel probabilities, $\Gamma\,dt=\sum_j\Gamma_j\,dt$ , hence $\Gamma=\sum_j\Gamma_j$ . The lifetime is $\tau=1/\Gamma=1/\sum_j\Gamma_j$ . Why: independent processes have additive rates — opening a new decay channel can only make a particle die faster, never slower, because it adds a fresh way to leave.

Step 4. The share of decays going through channel $j$ is its rate over the total, $\mathrm{BR}_j=\Gamma_j/\Gamma$ , and summing over all channels gives $\sum_j\mathrm{BR}_j=1$ . Why: every particle eventually decays through exactly one channel, so the fractions must partition unity — it is the branching ratios, not the partial widths, that sum to one. *Note: $\tau=1/\Gamma$ follows from the first-moment integral; opening channels increases $\Gamma$ and shortens $\tau$ ; $\sum\mathrm{BR}_j=1$ by construction.*

Limiting cases

Case	Result	Meaning
Massless daughters $m_1,m_2\to0$	$p^*\to m_a/2$	maximal phase space; the whole rest energy goes to kinetic momentum
Threshold $m_1+m_2\to m_a$	$p^*\to 0$ , so $\Gamma\to0$	no phase space — the decay switches off just above threshold
One dominant channel	$\Gamma\approx\Gamma_j$ , $\mathrm{BR}_j\approx1$	lifetime is set almost entirely by that mode (e.g. $\pi^\pm\to\mu\nu$ , 99.99%)
Many open channels	$\tau=1/\sum_j\Gamma_j$	the total width exceeds every partial width; more doors, shorter life
Heavy weak decay $m_a\gg m_f$	$\Gamma\propto G_F^2\,m_a^5$	Sargent's rule — the strong mass dependence behind short heavy-particle lifetimes
Near a resonance mass	width $\Gamma$ = peak full-width in the Breit–Wigner lineshape	for a state too short-lived to time, $\Gamma$ is read off the mass spectrum, not a clock

Common trapConfusing the partial width with the total width

The single most common slip is to think a large partial width $\Gamma_j$ by itself means a short lifetime, or to expect the partial widths to sum to one. Neither is right. The lifetime is fixed by the total width, $\tau=\hbar/\Gamma=\hbar/\sum_j\Gamma_j$ , so a channel with a big $\Gamma_j$ shortens the life only in proportion to its contribution to the sum. What sums to unity is the set of branching ratios, $\sum_j\mathrm{BR}_j=1$ , because every particle must decay through exactly one channel. Partial widths carry units of energy and can be any positive numbers; branching ratios are dimensionless fractions between 0 and 1. Keep the two ideas apart: partial widths add up to the total width, and it is only after dividing by that total that you get the fractions that add up to one.

Worked examples

1 · Pion lifetime, width, and phase-space momentum (fully worked)

The charged pion decays almost entirely by $\pi^+\to\mu^+\nu_\mu$ with mean lifetime $\tau=2.603\times10^{-8}$ s. Take $m_\pi=139.57$ MeV, $m_\mu=105.66$ MeV, $m_\nu\approx0$ . Find (a) the rest-frame momentum $p^*$ , (b) the daughter energies, and (c) the total width $\Gamma$ in eV.

Step 1. Phase-space momentum from (3) with $m_2=0$ , which reduces the Källén form to $p^*=(m_a^2-m_1^2)/(2m_a)$ . Numerically $m_\pi^2=19{,}480\ \mathrm{MeV}^2$ and $m_\mu^2=11{,}164\ \mathrm{MeV}^2$ , so $p^*=(19{,}480-11{,}164)/(2\times139.57)=8316/279.1=\mathbf{29.8\ MeV/c}$ .

Step 2. Daughter energies in the rest frame. The muon has $E_\mu=\sqrt{p^{*2}+m_\mu^2}=\sqrt{29.8^2+105.66^2}=\sqrt{888+11{,}164}=\mathbf{109.8\ MeV}$ , and the (massless) neutrino carries $E_\nu=p^*=\mathbf{29.8\ MeV}$ . Check energy conservation: $109.8+29.8=139.6\ \mathrm{MeV}=m_\pi$ . The books balance.

Step 3. Total width from the lifetime, using $\Gamma=\hbar/\tau$ with $\hbar=6.582\times10^{-16}$ eV·s. Then $\Gamma=(6.582\times10^{-16}\ \mathrm{eV\cdot s})/(2.603\times10^{-8}\ \mathrm{s})=\mathbf{2.53\times10^{-8}\ eV}=25.3\ \mathrm{neV}$ . A pion lives an eternity by particle standards, and its width is correspondingly minuscule — utterly unmeasurable as a linewidth, which is why the pion is timed, not weighed. Γ ≈ 25 neV, τ ≈ 26 ns.

2 · The Z boson: width, lifetime, and counting neutrinos

The $Z$ boson has total width $\Gamma_Z=2.495$ GeV. Its branching ratios are: hadrons 69.9%, charged leptons 10.1% (split equally over $e,\mu,\tau$ ), and invisible (neutrinos) 20.0%. Find (a) its lifetime, (b) the partial width to a single charged-lepton pair, and (c) how the invisible width counts neutrino generations.

Step 1. Lifetime from (4): $\tau=\hbar/\Gamma_Z=(6.582\times10^{-25}\ \mathrm{GeV\cdot s})/(2.495\ \mathrm{GeV})=\mathbf{2.64\times10^{-25}\ s}$ . Fourteen orders of magnitude shorter-lived than the pion, and far too brief to track — the $Z$ is characterized entirely by its width, measured as the breadth of the resonance peak.

Step 2. A single charged-lepton channel has $\mathrm{BR}=10.1\%/3=3.37\%$ , so $\Gamma_{\ell\ell}=0.0337\times2.495\ \mathrm{GeV}=\mathbf{84\ MeV}$ for each of $ee,\mu\mu,\tau\tau$ . The corresponding invisible partial width is $\Gamma_{\rm inv}=0.200\times2.495=\mathbf{499\ MeV}$ .

Step 3. Each neutrino species contributes a partial width $\Gamma_\nu\approx167$ MeV (the Standard Model value). Dividing, $N_\nu=\Gamma_{\rm inv}/\Gamma_\nu\approx499/167\approx3.0$ . Done precisely, LEP found $\mathbf{N_\nu=2.984\pm0.008}$ — decisive evidence for exactly three light neutrino generations. A branching ratio, read from a resonance width, settled one of the deepest questions about the structure of matter.

Physics of the portrait

Decay-rate theory is inseparable from one physicist. The formula that turns an interaction into a rate — dynamics times phase space — is Fermi's, born from his 1933 theory of beta decay, and the entire edifice of partial widths, branching ratios, and lifetimes is built on it.

Origins · 1 minEnrico Fermi and how this was discovered

Enrico Fermi

1901–1954 · the golden rule and the theory of beta decay

Enrico Fermi built the bridge between an interaction and a measurable decay rate. In 1933–34, confronted with the continuous energy spectrum of beta decay and Pauli's conjectured neutrino, he wrote down a theory of the weak interaction as a contact coupling of four fermions — the proton, neutron, electron, and neutrino currents meeting at a point — and from it computed the shape of the electron spectrum and the decay rate. To get a rate he needed the transition probability per unit time, which he expressed as the squared matrix element times the density of final states; he later called this the "Golden Rule No. 2" in his Chicago lecture notes, and the name stuck to the whole class of first-order rate formulas. The structure he found — a coupling constant (now $G_F$ , the Fermi constant) times a matrix element times phase space — is exactly the factorization that organizes this article, and his prediction that beta-decay rates scale as a high power of the released energy (Sargent's rule, $\Gamma\propto Q^5$ ) is the same mass dependence that makes heavy particles short-lived. Fermi was the rare physicist supreme in both theory and experiment: the same man who quantized the weak interaction also built the first nuclear reactor in 1942. His four-fermion theory was eventually revealed as the low-energy limit of $W$ -boson exchange, but the golden rule survived every revolution intact — it is still the first thing anyone writes down to compute a decay.

History

1927

— Dirac's time-dependent perturbation theory yields the transition rate per unit time

1930

— Pauli postulates the neutrino to rescue energy conservation in beta decay

1933

— Fermi formulates the theory of beta decay with a four-fermion contact coupling

1950

— Fermi names the master rate formula the "Golden Rule No. 2" in his lecture notes

1989

— LEP measures the Z lineshape width and fixes the number of light neutrinos at three

Going further

Texts. Thomson, Modern Particle Physics, §3.3–3.3.1 (golden rule, two-body phase space, the $p^*$ formula) · Griffiths, Introduction to Elementary Particles, 2nd ed., §6.1 and Appendix B.1 (decay rates and kinematics) · Halzen & Martin, Quarks and Leptons, §4.3 (phase space) · Peskin & Schroeder, An Introduction to Quantum Field Theory, §4.5 (cross sections and decay rates from the S-matrix) · Schwartz, Quantum Field Theory and the Standard Model, Ch. 5. Key numerical anchors: equation (2) follows from (1) by the two-body phase-space collapse; $p^*=(m_\pi^2-m_\mu^2)/2m_\pi=29.8$ MeV for the pion; $\Gamma_\pi=\hbar/\tau_\pi=25$ neV and $\tau_Z=\hbar/\Gamma_Z=2.6\times10^{-25}$ s; $\sum_j\mathrm{BR}_j=1$ and $\tau=1/\sum_j\Gamma_j$ . All lifetimes and widths ( $\tau_\mu=2.197\ \mu$ s, $\tau_\pi=26$ ns, $\Gamma_Z=2.495$ GeV, $N_\nu=2.984$ ) are standard PDG values.

Prerequisites. Fermi's Golden Rule · Lorentz-Invariant Phase Space · Special-Relativistic Kinematics. Leads to. Resonances and the Breit–Wigner Lineshape · Muon Decay · Interaction Cross Sections.

Ask the tutor: Ask me to walk through the two-body phase-space collapse line by line, or to compute $p^*$ , the width, and the branching ratios for any decay you name.

Why does a decay rate factorize into a matrix element and phase space?

Because the two pieces answer independent questions. The matrix element $\mathcal{M}_{fi}$ comes from the interaction — it says how strongly the initial state couples to a specific final configuration, and it is where all the coupling constants, propagators, and spin structure live. The phase space comes purely from kinematics — it counts how many final states are available consistent with energy and momentum conservation, and it depends only on the masses and the number of final particles, not on the force involved. The golden rule multiplies "how strong is the coupling" by "how much room is there to decay into." This is enormously useful in practice: you compute the phase space once for each final-state multiplicity and reuse it, so that a strong, electromagnetic, or weak decay into the same number of bodies shares identical kinematics and differs only in $\mathcal{M}_{fi}$ .

What exactly is p*, and why is it fixed by the masses alone?

For a two-body decay in the parent's rest frame, $p^*$ is the common magnitude of the two daughters' momenta — they must be equal and opposite because the parent had zero momentum. What is surprising is that the daughters have no freedom in their speed: energy and momentum conservation together are two constraints, and with only the momentum magnitude and one energy left to fix, the equations have a unique solution. Solving $m_a=\sqrt{p^{*2}+m_1^2}+\sqrt{p^{*2}+m_2^2}$ gives the Källén formula (3), which depends on nothing but the three masses. The daughters can choose only the direction they fly (the solid angle $d\Omega$ in the master formula); their momentum magnitude is dictated. This is special to two bodies — with three or more, the momenta share a continuous range of values, which is why a three-body decay produces a Dalitz plot rather than a single line.

Partial width, total width, branching ratio — how do they relate?

Each open decay channel $j$ has a partial width $\Gamma_j$ , which is the rate for that specific process and carries units of energy. The total width $\Gamma=\sum_j\Gamma_j$ is the sum over all channels; it is what sets the lifetime, $\tau=\hbar/\Gamma$ . The branching ratio $\mathrm{BR}_j=\Gamma_j/\Gamma$ is the dimensionless fraction of decays going through channel $j$ , and these fractions sum to one. A concrete grip: the $Z$ has a total width of 2.495 GeV; its partial width to $\mu^+\mu^-$ is about 84 MeV; and its branching ratio to $\mu^+\mu^-$ is $84/2495\approx3.4\%$ . Partial widths add up to the total; branching ratios are the shares of the total. Confusing the two — for instance expecting partial widths to sum to one — is the classic beginner's error.

Why is a particle's width quoted in energy units like MeV or GeV?

It follows from the time–energy uncertainty relation and natural units. A state that lives for a time $\tau$ cannot have a perfectly sharp energy; its energy is smeared by $\Delta E\sim\hbar/\tau=\hbar\Gamma$ . Setting $\hbar=1$ (natural units), the total rate $\Gamma$ is that energy spread — so "width" is literal: it is the full width at half maximum of the resonance peak in a plot of production rate versus energy. A long-lived particle has a narrow peak (small $\Gamma$ in energy); a short-lived one has a broad peak. For very short-lived states like the $Z$ or a hadronic resonance, you cannot possibly time the decay, so you measure the width of the peak instead and convert with $\tau=\hbar/\Gamma$ . For long-lived states like the muon or pion, you time the exponential decay and convert the other way. Width and lifetime are two windows onto the same number.

Is exponential decay exact, or just a good approximation?

For all practical purposes it is exact, and it follows from a single premise: the decay probability per unit time is constant, independent of the particle's age. A muon that has already lived a microsecond is no more "due" to decay than a fresh one — decay is memoryless. That constant fractional loss integrates to a pure exponential, $N(t)=N(0)e^{-\Gamma t}$ . There are deep, tiny corrections: quantum mechanics predicts non-exponential behaviour at extremely short times (the "Zeno" regime, where the survival probability is quadratic, not linear, in $t$ ) and at extremely long times (a power-law tail). Both have been glimpsed in special systems, but they are utterly negligible for particle lifetimes over any observable window. The exponential you measure in a detector is exponential to far more decimal places than you can check.

Does opening a new decay channel change the other channels' rates?

No — and this is the key to why widths add. Each partial width $\Gamma_j$ is computed from its own matrix element and phase space, independent of what other channels exist. Adding a new channel does not slow the existing ones; it simply provides an additional way for the particle to die, so the total rate $\Gamma=\sum_j\Gamma_j$ increases and the lifetime shortens. What does change is every branching ratio, because they are all normalized by the (now larger) total: $\mathrm{BR}_j=\Gamma_j/\Gamma$ drops for the pre-existing channels even though their $\Gamma_j$ are untouched. This is exactly why a heavy particle with many open channels, like the top quark, is so short-lived — it has an enormous number of ways to decay, each contributing its rate to a very large total width (the top's is about 1.4 GeV, giving a lifetime shorter than the time it takes to hadronize).